Sources of Superlinearity in Davenport-Schinzel Sequences

A {em generalized} Davenport-Schinzel sequence is one over a finite alphabet that contains no subsequences isomorphic to a fixed {em forbidden subsequence}. One of the fundamental problems in this area is bounding (asymptotically) the maximum length of such sequences. Following Klazar, let $Ex(sigma,n)$ be the maximum length of a sequence over an alphabet of size $n$ avoiding subsequences isomorphic to $sigma$. It has been proved that for every $sigma$, $Ex(sigma,n)$ is either linear or very close to linear; in particular it is $O(n2^{alpha(n)^{O(1)}})$, where $alpha$ is the inverse-Ackermann function and $O(1)$ depends on $sigma$. However, very little is known about the properties of $sigma$ that induce superlinearity of $Ex(sigma,n)$. In this paper we exhibit an infinite family of independent superlinear forbidden subsequences. To be specific, we show that there are 17 {em prototypical} superlinear forbidden subsequences, some of which can be made arbitrarily long through a simple padding operation. Perhaps the most novel part of our constructions is a new succinct code for representing superlinear forbidden subsequences.